What do you want (prefer) to do this evening? (evening here ~ countable noun)
To watch a movie.
What do you prefer doing at evening? (evening here ~ uncountable noun)
Watching film, having supper, then going out.
What do you like to do? (at the moment)
To stay at home.
What do you prefer, staying at home or going out?
Staying at home.
What do you prefer, to stay at home or to go out with that girl?
To go out, of course!
What options are correct?
Also I think Which option is correct?
is also possible if there are two options only and both are given.
What will be my answers talking face to face, I have no idea.